Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,23)(9,26)(10,27)(11,15)(12,19)(13,17)(14,25)(18,30)(20,33)(21,29)(22,36)(24,31)(28,35)(32,34);(5,7,17,34,10,22,9,21,11,20,24,18,25,35,8,19)(6,12,23,28,14,30,31,33,15,29,26,36,27,32,13,16);(7,18,20,34)(8,24)(9,17)(10,25)(12,28,29,36)(13,26)(14,27)(16,32,33,30)(19,22,21,35)(23,31)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,d^c=dhi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250703, #72250707, #72250729, … |
| maximal normal subgroup of | #72250703, #72250707, #72250729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56463);